$$$\cos{\left(x e^{3} \right)}$$$ 的积分

求$$$\int \cos{\left(x e^{3} \right)}\, dx$$$。

设$$$u=x e^{3}$$$。

则$$$du=\left(x e^{3}\right)^{\prime }dx = e^{3} dx$$$ (步骤见»)，并有$$$dx = \frac{du}{e^{3}}$$$。

该积分可以改写为

$${\color{red}{\int{\cos{\left(x e^{3} \right)} d x}}} = {\color{red}{\int{\frac{\cos{\left(u \right)}}{e^{3}} d u}}}$$

对 $$$c=e^{-3}$$$ 和 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$${\color{red}{\int{\frac{\cos{\left(u \right)}}{e^{3}} d u}}} = {\color{red}{\frac{\int{\cos{\left(u \right)} d u}}{e^{3}}}}$$

余弦函数的积分为 $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$：

$$\frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{e^{3}} = \frac{{\color{red}{\sin{\left(u \right)}}}}{e^{3}}$$

回忆一下 $$$u=x e^{3}$$$:

$$\frac{\sin{\left({\color{red}{u}} \right)}}{e^{3}} = \frac{\sin{\left({\color{red}{x e^{3}}} \right)}}{e^{3}}$$

因此，

$$\int{\cos{\left(x e^{3} \right)} d x} = \frac{\sin{\left(x e^{3} \right)}}{e^{3}}$$

加上积分常数：

$$\int{\cos{\left(x e^{3} \right)} d x} = \frac{\sin{\left(x e^{3} \right)}}{e^{3}}+C$$

$$$\int \cos{\left(x e^{3} \right)}\, dx = \frac{\sin{\left(x e^{3} \right)}}{e^{3}} + C$$$A